Cliquish function

In mathematics, the notion of a cliquish function is similar to, but weaker than, the notion of a continuous function and quasi-continuous function. All (quasi-)continuous functions are cliquish but the converse is not true in general.

Definition

Let $X$ be a topological space. A real-valued function $f:X\to \mathbb{ℝ}$ is cliquish at a point $x\in X$ if for any $ϵ>0$ and any open neighborhood $U$ of $x$ there is a non-empty open set $G\subset U$ such that

$|f(y)-f(z)|<ϵ\mathrm{\forall }y,z\in G$

Note that in the above definition, it is not necessary that $x\in G$.

Properties

• If $f:X\to \mathbb{ℝ}$ is (quasi-)continuous then $f$ is cliquish.
• If $f:X\to \mathbb{ℝ}$ and $g:X\to \mathbb{ℝ}$ are quasi-continuous, then $f+g$ is cliquish.
• If $f:X\to \mathbb{ℝ}$ is cliquish then $f$ is the sum of two quasi-continuous functions .

Example

Consider the function $f:\mathbb{ℝ}\to \mathbb{ℝ}$ defined by $f(x)=0$ whenever $x\le 0$ and $f(x)=1$ whenever $x>0$. Clearly f is continuous everywhere except at x=0, thus cliquish everywhere except (at most) at x=0. At x=0, take any open neighborhood U of x. Then there exists an open set $G\subset U$ such that $y,z<0\mathrm{\forall }y,z\in G$. Clearly this yields $|f(y)-f(z)|=0\mathrm{\forall }y\in G$ thus f is cliquish.

In contrast, the function $g:\mathbb{ℝ}\to \mathbb{ℝ}$ defined by $g(x)=0$ whenever $x$ is a rational number and $g(x)=1$ whenever $x$ is an irrational number is nowhere cliquish, since every nonempty open set $G$ contains some $y_1,y_2$ with $|g(y_1)-g(y_2)|=1$.

References

• Ján Borsík (2007–2008). "Points of Continuity, Quasi-continuity, cliquishness, and Upper and Lower Quasi-continuity". Real Analysis Exchange 33 (2): 339–350. http://www.projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.rae/1229619412&page=record. 
• T. Neubrunn (1988). "Quasi-continuity". Real Analysis Exchange 14 (2): 259–308. doi:10.2307/44151947.